Question

Write the proper numbers in the boxes.

(1) ${\left[{\left(-2\right)}^3\right]}^4={\left(-2\right)}^{\boxed{}\times \boxed{}}$
(2) ${\left[{\left(-\frac{12}{13}\right)}^2\right]}^5={\left(-\frac{12}{13}\right)}^{\boxed{}}$
(3) ${\left({10}^5\right)}^6=10{}^{\boxed{}}$
(4) ${\left[{\left(\frac{5}{3}\right)}^{10}\right]}^3={\left(\frac{5}{3}\right)}^{\boxed{}}$
(5) ${\left(x^5\right)}^7=x^{\boxed{}}$
(6) ${\left(m^4\right)}^4=m^{\boxed{}}$

Answer 1

(1) ${\left[{\left(-2\right)}^3\right]}^4={\left(-2\right)}^{\boxed{}\times \boxed{}}$
Consider ${\left[{\left(-2\right)}^3\right]}^4$
$={\left(-2\right)}^{\boxed{3}\times \boxed{4}}\left[{\left(a^m\right)}^n={\left(a\right)}^{m\times n}\right]$

(2) ${\left[{\left(-\frac{12}{13}\right)}^2\right]}^5={\left(-\frac{12}{13}\right)}^{\boxed{}}$
Consider ${\left[{\left(-\frac{12}{13}\right)}^2\right]}^5$
$$\begin{gathered} ={\left(-\frac{12}{13}\right)}^{2\times 5}\left[{\left(a^m\right)}^n={\left(a\right)}^{m\times n}\right] \\ ={\left(-\frac{12}{13}\right)}^{\boxed{10}} \end{gathered}$$

(3) ${\left({10}^5\right)}^6={10}^{\boxed{}}$
Consider ${\left({10}^5\right)}^6$
= 10^5×6 [ (a^m)ⁿ = (a)^m×n ]
$={10}^{\boxed{30}}$

(4) ${\left[{\left(\frac{5}{3}\right)}^{10}\right]}^3={\left(\frac{5}{3}\right)}^{\boxed{}}$
Consider ${\left[{\left(\frac{5}{3}\right)}^{10}\right]}^3$
$$\begin{gathered} ={\left(\frac{5}{3}\right)}^{10\times 3}\left[{\left(a^m\right)}^n={\left(a\right)}^{m\times n}\right] \\ ={\left(\frac{5}{3}\right)}^{\boxed{30}} \end{gathered}$$

(5) ${\left(x^5\right)}^7=x^{\boxed{}}$
Consider ${\left(x^5\right)}^7$
$$\begin{gathered} =x^{5\times 7}\left[{\left(a^m\right)}^n={\left(a\right)}^{m\times n}\right] \\ =x^{\boxed{35}} \end{gathered}$$

(6) ${\left(m^4\right)}^4=m^{\boxed{}}$
Consider ${\left(m^4\right)}^4$
= m^4×4 [ (a^m)ⁿ = (a)^m×n ]
$=m^{\boxed{16}}$